Routh–Hurwitz theorem

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In mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.

1 Notations
2 Statement
3 Routh–Hurwitz stability criterion
4 References
5 External links

[edit] Notations

Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary line (i.e. the line Z=ic where i is the imaginary unit and c is a real number). Let us define $P 0 (y)$ (a polynomial of degree n) and $P 1 (y)$ (a nonzero polynomial of degree strictly less than n) by $f (i y) = P 0 (y) + i P 1 (y)$ , respectively the real and imaginary parts of f on the imaginary line.

Furthermore, let us denote by:

p the number of roots of f in the left half-plane (taking into account multiplicities);
q the number of roots of f in the right half-plane (taking into account multiplicities);
$\Delta\arg f(iy)$ the variation of the argument of f(iy) when y runs from -∞ to +∞;
w(x) is the number of variations of the generalized Sturm chain obtained from $P 0 (y)$ and $P 1 (y)$ by applying the Euclidean algorithm;
$I_{-\infty}^{+\infty}r$ is the Cauchy index of the rational function r over the real line.

[edit] Statement

With the notations introduced above, the Routh–Hurwitz theorem states that:

$p-q=\frac{1}{\pi}\Delta\arg f(iy)=-I_{-\infty}^{+\infty}\frac{P_1(y)}{P_0(y)}=w(+\infty)-w(-\infty).$

From the first equality we can for instance conclude that when the variation of the argument of f(iy) is positive, then f(z) will have more roots to the left of the imaginary axis than to its right. The equality p-q=w(+∞)-w(-∞) can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is p+q and the w from the right member is the number of variations of a Sturm chain (while w refers to a generalized Sturm chain in the present theorem).

[edit] Routh–Hurwitz stability criterion

Main article: Routh-Hurwitz stability criterion

We can easily determine a stability criterion using this theorem as its is trivial that f(z) is Hurwitz stable iff p-q=n. We thus obtain conditions on the coefficients of f(z) by imposing w(+∞)=n and w(-∞)=0.

[edit] References

Hurwitz, A., "On the Conditions under which an Equation has only Roots with Negative Real Parts", Rpt. in Selected Papers on Mathematical Trends in Control Theory, Ed. R. T. Ballman et al. New York: Dover 1964
Routh, E. J., A Treatise on the Stability of a Given State of Motion. London: Macmillan, 1877. Rpt. in Stability of Motion, Ed. A. T. Fuller. London: Taylor & Francis, 1975
Gantmacher, F.R., Applications of the Theory of Matrices. Trans. J. L. Brenner et al. New York: Interscience, 1959